Journal of Water Resources Planning and Management · 2017. 12. 14. · Journal of Water Resources Planning and Management under a range of31 operating conditions. The degree to which

Journal of Water Resources Planning and Management

Reliability Indicators for Water Distribution System Design: 1

A Comparison 2

Stuart Atkinson; Raziyeh Farmani; Fayyaz A. Memon; David Butler 3

Abstract 4

When designing a water distribution system (WDS) it is imperative that the reliability of the 5

network is taken into consideration. It is possible to directly evaluate the reliability of a 6

WDS, although the calculation processes involved are computationally intensive and thus 7

undesirable for some state-of-the-art, iterative design approaches (such as optimisation). 8

Consequently, interest has recently grown in the use of reliability indicators, which are 9

simpler and faster to evaluate than direct reliability methods. 10

In this study, two existing reliability indicators, the Todini resilience index and entropy for 11

WDS, are examined by analysing their relationships with different sub-categories of 12

reliability, namely the mechanical (network tolerance to pipe failure) and hydraulic reliability 13

(network tolerance to demand change). The analysis is performed by generating comparable 14

solutions through optimisation of cost against the chosen indicators using the well known 15

Anytown WDS benchmark as a case study. 16

It is found that WDS solutions with high entropy have increased mechanical reliability, yet 17

are expensive and have poor hydraulic operation and water quality. In contrast, high 18

resilience index networks are relatively cheaper and present reasonable hydraulic operational 19

performance, yet have limited improvement in mechanical reliability. Both indicators appear 20

to be correlated to hydraulic reliability but each has its own associated disadvantages. When 21

optimised together, a trade-off between the two indicators is identified, inferring that 22

significantly increasing both simultaneously is not possible, and thus a new indicator is 23

recommended in order to account for both the mechanical and hydraulic reliability whilst 24

ensuring reasonable standards of hydraulic operation. 25

CE Database subject headings: Water Distribution Systems, Reliability, Rehabilitation, 26 Optimization, Pumps, Water Tanks, Water Quality 27

Introduction 28

Water distribution systems (WDS) are designed to provide consumers with a minimum 29

acceptable level of supply (in terms of pressure, availability and water quality) at all times 30


under a range of operating conditions. The degree to which the system is able to achieve this, 31

under both normal and abnormal conditions, is termed its reliability. An indication of system 32

reliability can in principle be calculated though the simulation of multiple system states under 33

an array of different network conditions and configurations (Maier et al. 2001). However, 34

this is likely to be computationally intensive and infeasible if optimal system solutions are 35

being sought. To overcome this limitation, various indicators have been developed that aim 36

to represent reliability yet do not have the computational requirements associated with the 37

direct analysis techniques (Baños et al. 2011). Ostfeld (2004) and Lansey (2006) reviewed a 38

number of definitions for reliability, spanning from simple topology or connectivity to more 39

complex definitions accounting for the hydraulic operation of a network and concluded that 40

each indicator has its strengths and weaknesses, but will typically only capture (to some 41

extent) the particular feature of reliability for which it was designed. 42

Reliability is typically sub-divided into two aspects. Mechanical reliability reflects the 43

degree to which the system can continue to provide adequate levels of service under 44

unplanned events such as component failure (e.g. pipe bursts, pump malfunction). Hydraulic 45

reliability reflects how well the system can cope with changes over time such as deterioration 46

of components or demand variations. Wagner et al. (1988) argued that both mechanical and 47

hydraulic reliability are important factors to consider during WDS design and both should be 48

accounted for explicitly. 49

Previous studies (Farmani et al. 2005; di Nardo et al. 2010; Raad et al. 2010) have examined 50

the extent to which key indicators (singly or in combination) are able to quantify both forms 51

of reliability (mechanical and hydraulic) within simple water distribution networks. This 52

paper presents a comprehensive, comparative analysis of popular reliability indicators based 53

on a more complex network containing pumps and tanks. The aim is to establish which 54

indicator, or combination of indicators, is able to accurately represent both the mechanical 55

and hydraulic reliability of a WDS, or whether a more comprehensive indicator is required. 56

Reliability Indicators 57

As mentioned a range of reliability indicators have been developed of various degrees of 58

sophistication. In general, these all give some indication of the ability of a WDS to cope with 59

changing conditions and are straightforward to calculate so are useful for optimisation studies 60

that compare the performance of one instance of a network design with another. None are 61


particularly significant as standalone values. This section presents the definition of the key 62

indicators and their derivatives, together with advantages and disadvantages where known. 63

Resilience Index 64 Todini’s resilience index is a popular surrogate measure within the WDS research field 65

(Todini 2000; Prasad and Park 2004; Farmani et al. 2005; Saldarriaga and Serna 2007; Reca 66

2008), which considers surplus hydraulic power as a proportion of available hydraulic power. 67

The resilience index, Ir, is measured in the continuous range [0…1] (for feasible solutions of 68

ha,i≥hr,i) and is formulated as (Todini 2000): 69

70

I =∑ q h , − h ,∉

∑ Q H + ∑∈ − ∑ q h ,∉

(1)

nn Number of supply and demand nodes np Number of pumps IN Set of supply nodes (reservoir/emptying tanks) ha,i Available head at supply node i (kPa) hr,i Required head at supply node i (kPa) qi Demand at node i (m3/s) Qi Supply at input node i (m3/s) Hi Head from input node i (kPa) Pj Power from pump j (kW) γ Specific weight of water (N/m3) 71

The resilience index has been shown to be correlated to hydraulic and to some extent 72

mechanical reliability (Farmani et al. 2005), yet the function has also been shown to exhibit 73

some weaknesses. Several adaptations of the resilience index have been developed in order to 74

account for (a) the degree of uniformity of pipe diameters entering nodes, i.e. the network 75

resilience (Prasad and Park 2004), and (b) to combat inconsistencies with the indicator when 76

considering multiple sources, i.e. the modified resilience index (Jayaram and Srinivasan 77

2008). Baños et al. (2011) compared the three indexes in a two objective (cost vs. reliability 78

indicator) study and revealed that there was some correlation between each resilience 79

indicator and hydraulic reliability but that the two newer indicators did not particularly 80

improve on the original. Indeed, with no overall ‘best’ indicator, it was suggested that all of 81


these resilience indicators are incapable of fully considering the connectivity of a network 82

and thus are unable to identify the most critical areas in systems requiring reinforcement. 83

Entropy 84 The entropy reliability indicator was first developed by Awumah et al. (1990) and later used 85

by Tanyimboh & Templeman (1993). It assesses the ‘disorder’ of flow around a network by 86

taking into account the proportions of flow entering individual nodes, thus providing a 87

surrogate measure of network connectivity (number of possible flow paths). Maximising 88

entropy has been shown to increase a network’s mechanical reliability (Awumah et al. 1990). 89

The maximum achievable entropy value has no standard range, and is dependent upon the 90

number of nodes within a network and the number of pipes attached to these. Tanyimboh 91

and Templeman’s (1993) formulation of entropy (S) is given in equation 2: 92

S = −QT ln

QT −

1T T

qT ln

qT +

qT ln

qT

∈∉∈

(2)

T Total network inflow from reservoir/tanks (m3/s) Ti Total flow reaching node i (m3/s) Ni Set of direct upstream nodes j connected to node i qij Flow rate in pipe ij (m3/s)

93

Setiadi et al. (2005) performed a comparative study between entropy and mechanical 94

reliability (operation of the network after pipe failure) concluding that the two have a strong 95

correlation despite having different methods of calculation. Further developments in entropy 96

have been made through examining its application to more advanced networks (e.g. multiple 97

sources with demands split between them (Yassin-Kassab et al. 1999)). 98

Minimum Surplus Head 99 In a WDS, Minimum surplus head, Is, is defined as the lowest nodal pressure difference 100

between the minimum required and observed pressure, formulated as: 101

퐼 = min ℎ , − ℎ , ; 푖 = 1, …푛푛 (3)


Farmani et al. (2005) found that increasing the minimum surplus head in addition to the 102

resilience index can improve the connectivity and thus mechanical reliability. It is not known 103

if this same conclusion is valid with respect to the entropy indicator. 104

Performance 105 Two recent studies have shed some light on the performance of the resilience index and 106

entropy. Di Nardo et al. (2010) concluded that the two measures provided different 107

information about network hydraulic behaviour. The resilience index was shown to be 108

strongly correlated with system pressure under failure conditions while entropy was revealed 109

to have no significant correlations with any hydraulic performance measure. Their study also 110

highlighted that entropy values were sensitive to minor changes in the structural layout of the 111

simple network test. 112

Raad et al. (2010) examined the relationship between resilience index, network resilience, 113

entropy, and a combination of resilience index and entropy with hydraulic and mechanical 114

reliability. Their research concluded that although the resilience index correlated more 115

significantly with both forms of reliability than the other indicators, it was less effective in 116

ensuring the good connectivity needed in effective WDS design (Walski 2001). It was 117

concluded that a combination of resilience index and entropy gave the best alternative to the 118

resilience index alone. 119

Method 120

Multi-objective design optimisation will be used to generate a wide range of comparable 121

WDS solutions (i.e. with similar costs but varying reliability indicator values) based on a 122

basic case study. WDS solutions associated with different indicator values will be compared 123

through analysis of cost-indicator trade-offs and network components identified that 124

contribute most to increasing the magnitude of the indicators. Relationships between the 125

optimisation objectives will be explored to understand whether and how they are correlated. 126

Finally, the performance of the various indicators will be evaluated in terms of their 127

effectiveness in promoting high mechanical and hydraulic reliability of the WDS solutions. 128

The indicator combinations will be used for multi-objective optimisation to generate a 129

selection of cost-benefit trade-off solutions: 130

A) Cost (CTOTAL) vs. Resilience Index (Ir) 131 B) Cost vs. Entropy (S) 132


C) Cost vs. Resilience Index vs. Minimum Surplus Head (Is) 133 D) Cost vs. Entropy vs. Minimum Surplus Head 134 E) Cost vs. Resilience Index vs. Entropy 135

Optimisation analysis for cases A-E will be performed using a WDS hydraulic simulation of 136

the Anytown network (see below) in EPANET2 (Rossman 2000) coupled with the NSGAII 137

genetic algorithm (Deb et al. 2000). 138

Case Study: Anytown 139 The widely used benchmark network Anytown is a reasonably complex WDS with 140

requirement for both pumping and storage tanks and is thus well suited to this comparative 141

study (Walski et al. 1987). The underperforming Anytown network requires rehabilitation 142

and expansion in order to meet new nodal demands while satisfying all constraints presented 143

in Table 1. The network re-design requires the selection of existing pipes for cleaning or 144

duplication, along with sizing and siting of new tanks and identification of an appropriate 145

pump schedule for normal-day operation. In this study, this gives an opportunity not merely 146

to design to the minimum level of network operation (lowest cost feasible network) but to 147

allow for additional operational benefit (through optimisation of the surrogate reliability 148

measures against cost) in order to make the WDS more reliable under uncertain conditions 149

(the extent of which is to be determined through in this study). This will allow generation of 150

solutions with differing values of the surrogate reliability measures that can be used for 151

comparison. The Anytown WDS layout is shown in Fig. 1. The network is divided into two 152

costing-zones; the city (bold lines) and suburban (thin lines), where rehabilitative actions 153

taken inside the city-zone are more costly to instigate. The total cost (CTOTAL) for 154

implementing the selected rehabilitation procedures for a given solution are calculated as the 155

sum of pipe costs (CPIPE), new tank costs (CTANK) and the net present value of pump 156

operational costs over a period of 20 years (CPUMP). Where: 157

퐶 = 퐿 푐 (퐷 ,푍 ,퐴 ) (4)

퐶 = 푐 (푉 ) (5)

퐶 =

1− 푎1− 푎 푐 퐸 (6)

Where 푎 = 158


퐶 = 퐶 + 퐶 + 퐶 (7)

nl Number of pipes Lm Length of pipe m (m) cp Unit length cost of pipe to perform action ($/m) Dm Diameter of pipe m (m) Zm Pipe zone of pipe m (city or suburbs) Am Action for pipe m (clean, duplicate or new) Vt Total volume of tank t (m3) ct Cost of tank t as a function of volume (see CWS for calculation) ce Unit energy cost ($/kWh) Ep Total energy used by pump p over 24h (kWh) n Investment period (yrs) r Rate of return (r=12%)

159

160

Fig. 1. Anytown Benchmark Network (Farmani et al. 2005) 161

The location-dependant unit-length costs for cleaning, duplicating and adding new pipes (8 162

discrete pipe diameters), cp(D,Z,A), new tank installation costs, ct(V), and unit energy costs 163

for pumping, ce, along with further definition of the benchmark, are available from CWS 164

(2004). A set of constraints used within the study (defining a feasible solution) are presented 165

in Table 1, including ensuring existing tanks are used to their full daily operational capacity 166

in addition to satisfying minimum individual nodal pressures for the five operational 167

scenarios (by by changing nodal demand to simulate peak flow and fire-flow conditions). 168

The variables used for optimisation, associated with the selection of new and duplicate pipes, 169

cleaned pipes, tank properties and pump scheduling, are given in Table 2. 170


Table 1. Design Constraints 171

Description Violation Condition

24h normal-day operation Any node < 276kPa

Instantaneous peak demand (1.8 times average demand) Any node < 276kPa

0.158m3/s (2500gpm) fire flow in node 19, DM of 1.3 at all other nodes Any node < 138kPa

0.095m3/s (1500gpm) fire flow in nodes 5,6 & 7, DM of 1.3 at all other nodes Any node < 138kPa

0.063m3/s (1000gpm) fire flow in nodes 11 & 17, DM of 1.3 at all other nodes Any node < 138kPa

Existing tanks use their full operational volume < 100%

Tank start level same as tank end level over 24h > 0m

Table 2. Design Variables 172

Description Range Number of variables

Tank maximum level relative to attached node 61.0-76.2m 2

Tank simulation start level 0-100% 4

Size of emergency storage (height below minimum

operating tank level)

0-7.6m 2

Diameter for new cylindrical tanks 1.5-30.5m 2

Level difference for normal day operation tank storage 0-15.2m 2

Locations of new tanks 0-32 2

Do nothing, clean an existing pipe or duplicate it 0-15 35

Assign discrete diameter to new pipe 0-15 8

Pump schedule for each time period of a 24h simulation 0-4 8

Results 173

General Performance 174 In order to understand which network components may influence or be influenced by the 175

reliability indicators, results from cases A-E were used to identify correlations (through 176

regression analysis) between the reliability indicators and the following: 177

Total network costs and cost breakdown (pipes, tanks and operation) 178

Minimum surplus head 179

Alternative indicator comparison (Resilience Index vs. Entropy) 180

Network Costs 181 Examination of the two-objective (total rehabilitation cost (Eq.7) vs. indicator) trade off 182

curves produced for cases A and B showed that the maximum resilience index for the 183

Anytown benchmark can be achieved at much lower total cost (CTOTAL) than that of the 184

maximum entropy. Cost was examined in more detail by breaking it down into components 185


(Eq. 4-6). The analysis showed that overall costs (CTOTAL) for both sets of reliability 186

indicators were strongly correlated (R2=0.998 in both cases) to network pipe cost (CPIPE). 187

However, for case A, an initial improvement of the resilience index appeared to be achievable 188

by altering pump scheduling and tank properties whilst maintaining consistent piping 189

expenditure (Fig. 2a). In contrast, case B (entropy) was mostly dependant on pipe costs, 190

which followed a linear path. For operational pumping cost (CPUMP), a moderate negative 191

correlation (R2=0.51) was noted against overall cost in case B suggesting that higher cost 192

entropy solutions have reduced operational cost. On inspection of Fig. 2b, the pumping 193

operational cost data for solutions was divided into several “clusters,” for which the 194

optimised tank locations were deemed as a possible cause (each cluster could be attributed to 195

separate new tank locations). 196

The overall cost of resilience index solutions (CTOTAL) presented limited correlation 197

(R2=0.171) with respect to tank cost (CTANK) (Fig. 2c). This is most likely because tank cost 198

is directly related to volume rather than height, operation or location, which necessitate 199

additional pumping capacity and thus are instead most likely reflected in operational cost 200

(CPUMP). In contrast, the entropy index presented a reasonable correlation against tank cost 201

(R2=0.7), although arguably this could be attributed to the weighting influence of the 202

previously identified location-dependant clusters. 203

Minimum Surplus Head 204 The influence of minimum surplus head was also investigated. The results from case A (Fig. 205

3a) show a positive correlation between the resilience index and minimum surplus head 206

(R2=0.94). However, case C (Fig. 3a) shows that the level of minimum surplus head can be 207

further increased for most resilience index values if considered together. For entropy, a weak 208

negative correlation (R2=0.39) was noted against minimum surplus head (Fig. 3b). In a 209

similar manner to the resilience index, there is potential to increase the minimum surplus 210

head for different entropy values if optimised together (Case D). This suggests for both cases 211

that the inclusion of minimum surplus head as a third objective should allow identification of 212

more valuable network solutions at equivalent cost. This conclusion, at least for the case of 213

resilience index, is supported by Farmani et al. (2005). 214


215

216

217

Fig. 2. Cost breakdown for solutions; Cases A & B (a) Total pipe costs (for new, clean and 218 duplicated pipes) (b) Pumping energy costs (NPV over 20 years) (c) New tank installation 219

costs 220

3

3.5

4

4.5

5

5.5

0.15

0.16

0.17

0.18

0.19

0.2

0.21

4 6 8 10 12

Entr

opy (

Case

B)

Resi

lienc

e In

dex

(Cas

e A)

Pipe Cost (CPIPE), $M

Case A Case B

3

3.5

4

4.5

5

5.5

0.15

0.16

0.17

0.18

0.19

0.2

0.21

6 6.1 6.2 6.3 6.4 6.5

Entr

opy (

Case

B)

Resi

lienc

e In

dex

(Cas

e A)

Operational Cost (CPUMP), $M

Case A Case B

3

3.5

4

4.5

5

5.5

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.5 0.6 0.7 0.8 0.9

Entr

opy (

Case

B)

Resi

lienc

e In

dex

(Cas

e A)

Tank Cost (CTANK,), $M

Case A Case B

(a)

(b)

(c)


221

222

Fig. 3. Cost vs. Minimum Surplus Head Relationship (A-D) (a) Minimum surplus head for 223 resilience index solutions (b) Minimum surplus head for entropy solutions 224

Alternative Indicator Comparison: Resilience Index vs. Entropy 225 In a similar manner to the minimum surplus head test, the relationship between the resilience 226

index and entropy of optimised solutions was also investigated. Fig. 4 indicates no 227

correlation for case A (R2=0.067) and a weak positive correlation (R2=0.356) for case B; yet 228

data for case B was clustered (clusters again related to separate tank locations). This implies 229

that optimising for either indicator individually will not necessarily achieve a high value of 230

the other indicator and simultaneous consideration (as in case E) may be necessary to 231

improve both. 232

Examination of the trade off between entropy and resilience index for the Anytown network 233

provides a clearer picture as to the interactions between the two indicators. Case E (where 234

both resilience index and entropy are optimised) in Fig. 4 clearly shows a maximum 235

resilience index after a certain level of entropy (S=3.74) is achieved. This suggests that there 236

0

0.5

1

1.5

2

0.14 0.16 0.18 0.2 0.22

Min

imum

Sur

plus

Hea

d (I S

) , m

Resilience Index (Ir)

C

00.20.4

0.60.8

1

1.21.41.6

1.8

3 3.5 4 4.5 5 5.5

Min

imum

Sur

plus

Hea

d (I s

), m

Entropy (S)

Case B Case D

(a)

(b)


is a considerable trade off between the two if higher entropy is desired. A similar shape can 237

also be noted in Fig. 3b (case D) for entropy and minimum surplus head. 238

239 Fig. 4. Entropy vs. Resilience Index Relationship Analysis (cases A, B & E) 240

Network Layout and Operation 241 This section focuses on identifying the extent to which the indicators improve the hydraulic 242

operation and reliability of the WDSs. This exercise is clearly important as the identified 243

trade-off between the resilience index and entropy means that it is unlikely that both can be 244

maximised simultaneously and therefore the reliability benefits from each will most likely 245

require trade-off. 246

Selected optimised solutions for cases A-E were considered for network level analysis to 247

identify which reliability indicator combinations were correlated to more desirable network 248

layout and operational features in terms of new pipe distribution (related to connectivity) and 249

hydraulic operation (in terms of pump scheduling and tank operation). Individual solutions 250

were selected systematically from the case A-E pareto-sets with the intention of providing a 251

range of indicator levels, while maintaining a similar cost for comparison between cases 252

(Table 3). This table provides a breakdown of information for each of the solutions 253

considered in this section. 254

On examination of the network layouts, it was noted that networks with mid-value resilience 255

indices (in cases A and C) appear to have duplicated pipes resembling a branched network 256

(Fig. 5a). This most likely ensures additional flow reaches each node with minimum 257

0.1

0.12

0.14

0.16

0.18

0.2

0.22

2 3 4 5 6

Resi

lienc

e In

dexb

(Ir)

Entropy (S)

Case A

Case B

Case E


expenditure. In contrast the high resilience index solutions appear to exhibit additional looped 258

zones reinforcing supply to nodes furthest from the source (Fig. 5a). 259

260

261

262

263

Fig. 5. Network example layouts for selected solutions (Both approx CTOTAL= $14.5M) (a) 264 Resilience index solution (C3), (b) Entropy solution (D2). 265

(a)

(b)

Storage Tank (O=original, N=New) Original Pipe (suburban/city) Duplicated Pipe (XXmmD=Duplicated pipe diameter) New Pipe (XXmm=New pipe diameter)

TO1

TN1

TO2

TO1

TO2 TN2

TN1

TN1


Examining the networks for cases B and D it is evident that increasing entropy solutions 266

exhibit an even distribution of duplicated pipes and thus a consistently increasing overall 267

system capacity (relative to solution cost) (Fig. 5b). This seems to have a generally negative 268

effect on the maximum water age for the networks (probably due to decreasing network 269

velocity and an increased number of available paths to each node), which is further magnified 270

with an increased minimum surplus head (case D). 271

The locations of new tanks are fairly consistent among the mid to high resilience index-based 272

solutions, both in case A, and even more so in case C. In contrast, new tank locations in 273

entropy solutions are more variable. Furthermore, the entropy indicator is not formulated to 274

directly consider tank operation, and indeed this has been apparent through notably poor tank 275

sizing in entropy solutions; in many cases increasing average storage time. Consequently, it 276

is plausible that the new tanks within (optimised) high entropy networks also add to the 277

problem of water aging (Table 3). 278

Examining system operation, it was noted that higher resilience index solutions generally 279

have higher new tank elevations (Fig. 6a) than the majority of those within high entropy 280

solutions (Fig. 6b). High entropy network tanks are also empty for extended periods of time 281

which could be problematic for both water aging and uncertain changes in demand as there is 282

consequently limited additional volume available. 283

Table 3. Cases A-E: Parameters for selected solutions (Objective in gray) 284

Case ID Is(m) Ir S Max Age (hours)

Solution cost breakdown ($M) Pipes Tanks Operation Total

A1 1.13 0.18 2.9 39.1 4.81 0.6 6.00 11.15 A2 1.26 0.19 2.91 41.8 6.11 0.59 6.05 12.75 A3 1.37 0.21 2.91 42.7 7.62 0.59 6.27 14.47 B1 1.01 0.13 3.26 39 5.86 0.63 6.23 12.71 B2 0.95 0.13 3.83 39 6.22 0.62 6.18 13.01 B3 0.25 0.15 4.66 49 7.68 0.76 6.12 14.56 B4 0.19 0.15 5.30 48 9.72 0.78 6.12 16.62 C1 1.06 0.17 2.61 40 4.74 0.68 6.18 11.62 C2 1.46 0.18 2.47 37 5.96 0.69 6.17 12.82 C3 1.75 0.20 2.52 44 7.69 0.68 6.16 14.52 D1 1.16 0.14 3.86 51 6.36 0.66 6.14 13.14 D2 1.36 0.15 4.34 66 7.72 0.74 6.20 14.67 D3 1.02 0.16 4.57 53 8.53 0.71 6.11 15.36 D4 0.71 0.16 5.08 84 13.0 0.77 6.25 20.06 E1 1.11 0.17 3.79 42 6.81 0.63 6.13 13.57 E2 1.04 0.16 9.96 47 7.96 0.59 6.34 14.88 E3 0.58 0.18 4.53 42 8.69 0.98 6.21 15.88 E4 0.66 0.17 5.01 61 11.5 0.98 6.04 19.01 285


286

287

288

Fig. 6. Tank Levels & Pump Scheduling for Solutions (a) C3 and (b) D2 (with (c) the 289 average 24h demand profile). Refer to Fig. 5 for tank labelling 290

Mechanical Reliability 291 The correlation between the reliability indicators and mechanical reliability was next 292

considered. A similar approach to that developed by Farmani et al. (2005) was used to 293

examine the effects of individual pipe failure against the available level of supply. Pipes 294

were closed individually and the fixed network was hydraulically simulated for a 24h 295

average-day operational demand profile (see Fig. 6c). The first hourly time period at which 296

hydraulic failure (pressure deficiency) occurred was noted and the next pipe in the series 297

assessed. If the failure time was in excess of 24h, the pipe was ignored within the simulation, 298

as major pipe failures are expected to be repaired within a day. Table 4 shows the results 299

from the mechanical reliability assessment (cumulative pipes that cause failure over 24hrs) 300

for the selected solutions investigated in section 4.2. 301

Examination of the results in Table 4 indicates that the resilience index in case A solutions 302

showed limited correlation to total pipes causing pressure failure over 24h. In contrast, case 303

B solutions demonstrated a gradual improvement with increasing entropy. This could be 304

explained by the notion that resilience index (case A) considers the average performance of 305

the network and localised issues (at individual nodes/zones) may not be captured. For 306

increasing entropy, an improvement is unsurprising, as the indicator promotes extra capacity 307

within networks. 308

64

68

72

76

0:00:00 12:00:00 24:00:00

Tank

Leve

l, m

Time (am-pm)

O1 O2 N1

64

68

72

76

0:00:00 12:00:00 24:00:00

Tank

Leve

l, m

Time (am-pm)

O1 O2 N1 N2

0123

Pum

ps

0123

Pum

ps

0

1

2

Dem

and

(a) (b)

(c)


A notable improvement in correlation between the indicators and pipes that caused failure 309

over 24h was observed through failure testing of the selected networks both for case C and D. 310

Of these, a considerable improvement was noted in case D, which at the maximum level of 311

entropy resulted in no failures over the 24 hour testing period for any single pipe out of 312

action. Although for this case, the cost of designing to the maximum level of entropy (which 313

exhibited the best mechanical reliability) was almost double that of the minimum cost 314

feasible network solution. 315

Case E demonstrated a reasonable compromise for the two sets of indicators, with mechanical 316

reliability not necessarily as high as observed in case D, but an improvement on high 317

resilience index only networks. Although the utilisation of a combination of indicators (as in 318

case E) was also deemed a reasonable compromise by Raad et al. (2010), the results for this 319

section indicated some differences to this previous work, as it was identified that the 320

resilience index exhibited improved mechanical reliability as compared with entropy. This 321

suggests that either the consideration of minimum surplus head or additional WDS 322

components (as in this study) may alter the correlation with mechanical reliability for both 323

surrogate reliability measures. 324

Table 4. Cases A-E: Results for mechanical reliability assessment: cumulative pipes that 325 cause pressure failure 326

Case ID Failure Test Results (hours to failure) 15 16 17 18 19 20 21 22 23 24

A1 4 6 6 6 6 6 6 6 6 6 A2 5 6 6 6 6 6 6 6 6 6 A3 3 6 6 6 6 6 6 6 6 6 B1 3 4 5 5 5 5 5 5 5 5 B2 3 4 4 4 4 4 4 4 4 4 B3 2 2 2 3 3 3 3 3 3 3 B4 0 0 0 1 1 1 1 1 1 1 C1 3 3 3 4 4 4 4 4 4 4 C2 3 4 4 4 4 4 4 4 4 4 C3 2 3 3 3 3 3 3 3 3 3 D1 2 2 2 2 2 2 2 2 2 2 D2 1 1 1 1 1 1 1 1 1 1 D3 1 1 1 1 1 1 1 1 1 1 D4 0 0 0 0 0 0 0 0 0 0 E1 1 1 1 1 2 2 2 2 2 2 E2 0 0 1 1 1 1 1 1 1 1 E3 0 1 1 1 1 1 1 1 1 1 E4 0 1 1 1 1 1 1 1 1 1

327


Hydraulic Reliability 328 Hydraulic reliability was evaluated by calculating the maximum average daily demand that a 329

given WDS solution is able to tolerate whilst maintaining feasible operation. This method is 330

used to represent a network in the future, when pump scheduling is a low cost option to alter 331

the hydraulic operation without costly or invasive rehabilitation procedures. For this 332

assessment, the pumping is optimised for each systematic change in demand to find if the 333

network is able to operate feasibly (with respect to minimum pressure and tank operation) 334

under these new demand conditions (further detail of this procedure is presented in Atkinson 335

et al.(2011). 336

Results from the analysis showed that both the resilience index and entropy alone (cases A 337

and B) presented limited correlation with hydraulic reliability. This could be attributed to 338

limited surplus head at underperforming nodes (which are not directly considered within 339

either indicator). With the additional improvement of minimum surplus head (in cases C and 340

D) a major improvement in correlation with the hydraulic reliability was noted. Case C 341

solutions revealed a positive relationship against hydraulic reliability; with the improvement 342

most likely due to the combination of new tank elevations (higher than entropy solutions) and 343

additional minimum surplus head. In contrast, the high tank elevations previously attributed 344

to more expensive case C solutions also appeared constraining for higher future demands (the 345

networks were unable to provide enough head to fill new tanks due to increased head-loss 346

when attempting to meet higher demands). This resulted in a capping effect in high resilience 347

index networks, where the maximum achievable demand is restricted (in the case of Anytown 348

it was found to be capped at around a 20% demand increase), and thus additional capital 349

expenditure was required in order to facilitate further demand increase. Case D solutions 350

revealed a positive correlation between network cost and hydraulic reliability although it was 351

more expensive to achieve similar hydraulic reliability levels in comparison to that observed 352

with case C. Nevertheless, a proportion of higher costing case D solutions outperformed any 353

other case solutions investigated under this category with a tolerance of up to a 25% increase 354

in demand, most likely due to the reduced system head-loss, and therefore more effective 355

pump operation (Atkinson et al. 2011). It is therefore difficult to distinguish whether case C 356

or D could be deemed more beneficial for improving hydraulic reliability, with the resilience 357

index showing a sharp but capped improvement in hydraulic reliability (against cost) 358

compared to a steady but less constrained improvement as observed within entropy solutions. 359


Conclusion 360

A comparison was conducted between two popular WDS reliability indicators. Comparable 361

WDS solutions, with respect to cost, were generated through optimisation of the Anytown 362

case study for each indicator (both individually and combined). The resultant solutions were 363

compared with respect to their ability to tolerate pipe failure (mechanical reliability) and 364

change in demand (hydraulic reliability), along with examination of the technical quality of 365

hydraulic operation. 366

It was found that networks with increased minimum surplus head alongside the reliability 367

indicators had generally improved all round performance in all tests performed. Solutions 368

with high entropy had notably improved mechanical reliability, while the resilience index 369

solutions were influenced to a lesser extent. Both indicators showed an improvement in 370

hydraulic reliability for higher magnitude solutions, although there was identification of a 371

trade-off between the relatively cheaper resilience index networks (limited to a maximum 372

redundant capacity) and the more expensive (but less limited capacity) high entropy 373

networks. In terms of hydraulic operation, the majority of the resilience index solutions 374

showed more desirable performance in terms of storage tank operation and the average 375

system water age (which was in many cases unacceptable in high entropy solutions). 376

For the case that the resilience index and entropy were optimised together, the performance 377

of resultant WDS networks over all testing categories was reasonable but could not easily be 378

accounted to either indicator individually. For this reason, and the significant observation that 379

there was considerable trade-off between the resilience index and entropy for higher cost 380

solutions, it is suggested that a new indicator is required that is able to measure/influence 381

both the connectivity and demand capacity of a WDS whilst also accounting for the quality of 382

hydraulic operation and water ageing. 383

Acknowledgements 384

This work was supported by the UK Engineering and Physical Science Research Council as 385

part of the Urban Futures Project (EP/F007426/1). 386


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